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Division by zero conjecture
The division by zero conjecture was initially proposed on 25 August 2016 to become a strong no-go theorem that asserted 0 could never have a multiplicative inverse in all conceivable algebraic structures. ''In other words, no nontrivial division by zero could exist without causing contradictions. On 13 October 2016, however, a set of counterexamples to this conjecture was found, thus showing that nontrivial division by zero algebra, with a true multiplicative inverse of zero, is actually possible, at the cost of losing the inductive axiom 1+1=2 and often associativity. The disproval of this conjecture opens up the exploration of the nature of zero terms in binary zero term algebras, where it is believed the properties of zero terms can be exploited in the future. Preliminary 1. For two binary operators \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}\# and \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white} \circ in some algebraic structure \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}S to be defined multiplication and addition respectively, at least a one sided distributive law must exists to relate them, that is for all \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a,b,c \in S \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a \# (b\circ c)=a \# b \circ a \# c and/or \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white} (a\circ b)\# c=a \# c \circ b \# c 2. An element \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0 in some algebraic structure \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}S with some binary operator \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}* is called a '''zero element' if at least one of the following is true: * Absorber: For all \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a\in S , \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a=a*0=0 * Identity: For all \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a\in S , \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a=a*0=a 3. Division by zero asserts there exists an element \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}q\in S such that \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}q*0=0*q=1 , where \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}1 is the multiplicative identity. Argument for the conjecture Consider \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}S , with 0, 1, q and left distributive law holds 1. It can be easily shown that \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white} \begin{align}0 & = 0\# 1 & \text{(Right multiplicative identity)}\\ & = 0 \# (1 \circ 0)& \text{(Left additive identity)}\\ & = 0 \# 1 \circ 0 \# 0 & \text{(Left distributivity)}\\ & = 0 \circ 0 \# 0 & \text{(Right multiplicative identity)}\\ & = 0 \# 0 & \text{(Left additive identity)}\\ \end{align} Therefore, 0 absorbs itself. 2. Since 0 has an inverse, it cannot be an absorber in general, otherwise operating \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}q , regardless of whether it is a one sided multiplicative inverse of 0, the trivial ring condition 0=1 can be shown. 3. Distributivity cannot be relaxed as the absence of distributive law results in the inability to tell which operator is multiplication and which is addition. In addition, additive identity cannot be discarded else S has no zero elements. Multiplicative identity is required in the definition of the multiplicative inverse q. 3. Therefore there exists no element 0 that is a zero element such that a multiplicative inverse exists. 4. Therefore division by zero does not exist for binary operator algebraic structures. As the counterexample demonstrated, the distributive law, identities and inverses interact in a nontrivial fashion, thus this worst case scenario argument cannot be generalised to all cases, thus the counterexamples were not ruled out by this argument. The status of the conjecture for non binary operators It is known that Wheels allow division to be always well defined. However it was shown that /0,0,1,0/0 are not inverses of zero and hence zero has no inverse. If /0 is the inverse of zero then 0/0=1 which is proven to result in the trivial wheel. Thus the conjecture holds for wheels. Category:Division by zero